5.1. Two-Dimensional Green–Ampt Problem
The application of the infiltration problem in unsaturated soil is examined. The governing equation used to describe water infiltration into unsaturated soils is the Green–Ampt equation. This equation relies on simplifying assumptions that make the complex infiltration process more manageable to describe.
This study focuses on the Green–Ampt problem, with the governing equation represented as Equation (4).
Figure 8 depicts the arrangement of collocation points in unsaturated soil with specified dimensions of 1 m in length (
a) and 1 m in height (
L). The Green–Ampt scenario simulates the infiltration of rainwater into unsaturated soil. Initially, the unsaturated soil is dry, and as time progresses, rainwater gradually infiltrates the unsaturated soil. The analysis evaluates the distribution of unsaturated flow. To simulate this infiltration behavior, the boundary conditions are defined as follows: the top of the unsaturated soil is set as a constant head boundary with a pressure head of zero. The left, right, and bottom boundaries are set to a dry condition with a pressure head of −1. This setup models the infiltration process from the top into unsaturated soil, where the infiltration rate quickly diminishes to reach a dry state.
The exact solution applied to model the aforementioned infiltration scenario is expressed as follows [
35,
36]:
where
represents the transient linearized pressure head,
L represents the height of the unsaturated soil,
represents the dry pressure head, which is set to 1,
a represents the length of the unsaturated soil,
k is the non-negative integer,
is a known value defined as
,
is the characteristic length,
is a known value defined as
is a known value defined as
,
is a known value defined as
,
i represents the positive integer, and
represents the characteristic length. In this case,
,
, and
. The unsaturated soil parameters, including
,
,
, and
, are 2 × 10
−5, 10
−4, 0.35, and 0.14, respectively. The total simulation time is one hour.
Finally, the linearized pressure head can be converted to the actual pressure head using the following equation:
The initial condition is set by placing boundary points along the edges and bottom of the three-dimensional spacetime coordinate system. Source points, located outside this cubic region, are arranged in a scattered, spiral manner. For solving the two-dimensional Green–Ampt problem using the proposed method, the configuration includes 119 to 137 source points and 1070 boundary points. Previous validation results showed that combining a DNN with spacetime MQ RBF provided the highest accuracy; hence, this application exclusively utilizes the proposed approach for analysis.
In this study, the proposed spacetime RBF-based DNNs architecture divides the dataset into three parts: 70% is allocated for training, while the remaining 30% is split equally between validation and testing, with each subset comprising 15% of the data for analysis. The training dataset consists of 193 samples, while both the validation and test datasets each contain 65 samples. To evaluate the distribution of unsaturated flow over time, this study employed 4056 uniformly spaced internal points within the cubic spacetime domain to represent pressure head distributions at different time intervals.
Figure 9 demonstrates the performance for the two-dimensional Green-Ampt equation. From the results of the training state of the proposed approach, as shown in
Figure 9a, the iteration count was configured to a maximum of 1000, with termination occurring at 853 iterations. The optimal validation mean squared error was 5.96 × 10
−14, the optimal test mean squared error was 7.53 × 10
−14, and the optimal training mean squared error was 7.31 × 10
−15.
Moreover, the results of the neural network training regression, depicted in
Figure 9b, demonstrate that the pressure head distributions derived from the proposed method with spacetime MQ RBF align closely with the analytical solution provided by Equation (21). Error metrics reveal that the RMSE, NSE, MRE, and MAE values are on the order of 10
−2, 10
−2, 10
−3, and 10
−1, respectively. These results highlight the high accuracy of the proposed method, showcasing its effectiveness in addressing the Green–Ampt problem. The findings confirm that the proposed spacetime RBF-based DNNs offer a highly accurate solution for the two-dimensional Green–Ampt problem.
5.2. Two-Dimensional Inverse Richards Equation
The inverse Richards equation involves determining the soil hydraulic properties or initial conditions from known measurements of pressure head or moisture content over time and space. In the context of unsaturated flow, this typically means inferring properties such as the soil’s hydraulic conductivity and moisture retention characteristics. Inverse problems are often ill-posed, meaning they can have multiple solutions or be sensitive to measurement errors, with small changes in input data leading to significant variations in the estimated parameters. Additionally, the accuracy of the inverse solution is highly dependent on the initial and boundary conditions used in the simulation. Incorrect assumptions or data can result in erroneous estimates.
The inverse Richards problem aims to reconstruct the historical distribution of unsaturated pressure head from current or final data. This problem is represented by Equation (4), and the computational domain is described by Equation (10), where
. In this study, the parameters including
,
,
,
, and
are 2 × 10
−5, 10
−4, 0.35, 0.14, and 5 respectively. The entire simulation runs for 5 h. The spacetime boundary condition in this scenario is specified by the analytical solution [
34] as follows:
This study considers four different cases. The differences between these cases lie in the range of given boundary conditions. In Case A, both the initial conditions and the full set of boundary conditions are specified. In Case B, only the boundary conditions over time are given, while the initial conditions are unknown. In Case C, only partial information is provided for both initial and boundary conditions. In Case D, partial boundary conditions are considered, and the initial conditions are unknown. Based on the scenarios described above, this study places boundary points according to the given boundary conditions, as shown in
Figure 10.
For Case A, Case B, Case C, and Case D, the boundary points number 1582, 1410, 766, and 720, respectively, and these points are evenly distributed according to the positions shown in
Figure 10. Additionally, the number of source points in all four cases ranges from 645 to 650. These source points are irregularly distributed outside the boundary points. This study employs a DNN architecture with spacetime RBF to simulate the inverse Richards equation. The proposed method divides the data into three segments: 70% for training, 15% for validation, and 15% for testing.
The network features two hidden layers optimized using the LM algorithm. The training process is set to run for 10
3 epochs, targeting a performance goal of 0 and a minimum performance gradient of 10
−7. The initial damping parameter is 10
−3, with an update strategy involving an increase factor of 10 and a decrease factor of 10
−1, capped at a maximum damping parameter of 10
10. Relevant parameters are detailed in
Table 1.
Furthermore, hyperparameters such as the activation function, loss function, optimization function, and hidden layer quantity are selected based on the hyperparameter tuning and comparison analysis presented in
Section 4. Moreover, evaluating the robustness of the proposed method for solving the inverse Richards equation requires examining its performance with data affected by random noise. This study performs such an assessment using noise-perturbed data.
The noise level refers to the degree of random variation or disturbance introduced to ideal or exact data. In machine learning, noise represents errors or uncertainties present in the input data or output results. It is often introduced to simulate real-world conditions where data is inherently imperfect. Moreover, noise level also refers to the intensity or power of the noise in a signal. It can be measured in terms of power or amplitude. For power, it is denoted as , and for amplitude, it can be expressed as the standard deviation of the noise signal. Furthermore, noise is typically quantified using either the standard deviation or the signal-to-noise ratio (SNR). The SNR measures the ratio between signal power and noise power. Additionally, the SNR is inversely proportional to the noise power; an increase in noise power results in a decrease in SNR, signifying a higher relative noise level compared to the signal. This inverse relationship highlights how increasing noise levels degrade the clarity and quality of the signal.
To solve the inverse Richards equation, it is essential to assess the effectiveness and stability of the proposed numerical method. In this case, boundary and final time data affected by random noise are utilized. The noisy data applied to the accessible boundary and final time are as follows [
37,
38]:
where
represents the exact boundary data in Equation (4),
represents the exact final time data,
represents the noisy data on the accessible boundary,
represents the noisy data in the final time data,
s represents the level of noise, and
represents a random number in the range [0, 1] generated by the uniform distribution. The noisy data represented in Equations (24) and (25) are used to calculate the inverse Richards equation.
Table 6 presents the computed errors for the inverse Richards equation across 20 runs. The results show that Case A yields RMSE, MRE, and MAE values of 10
−6, 10
−7, and 10
−6, respectively. Case B produces RMSE, MRE, and MAE values of 10
−6, 10
−6, and 10
−5. Case C results in RMSE, MRE, and MAE values of 10
−6, 10
−6, and 10
−5. Case D shows RMSE, MRE, and MAE values of 10
−5, 10
−4, and 10
−4. The analysis of these cases suggests that the proposed method, developed in this study, can accurately determine the distribution of unsaturated flow in soils, even when boundary conditions are insufficient, as illustrated in
Figure 11. Furthermore, with the effect of noise at a level of 0.1, all four cases achieve RMSE, MRE, and MAE values of 10
−4, 10
−4, and 10
−3, respectively. This demonstrates that the proposed method developed in this study can effectively determine the distribution of unsaturated flow in soils, even when boundary conditions are inadequate and data is affected by random noise, as illustrated in
Figure 12. Although the results in
Table 6 show that when the boundary conditions are disturbed by noise, the error is lower compared to when there is no noise interference, the proposed DNN in this study still achieves reliable numerical accuracy. Furthermore, with noise at a level of 0.1, all four cases reach RMSE, MRE, and MAE values of 10
−4, 10
−4, and 10
−3, respectively. This demonstrates that the proposed DNN in this study can effectively determine the distribution of unsaturated flow in soils, even when boundary conditions are inadequate and data is affected by random noise, as shown in
Table 6.